where \(\kappa _{0}\)  is the enhancement factor for the given alkali metal and noble gas pairs (for example, for \(^3\) He and K, \(\kappa _{0} \approx 5.9\) ),
\(g\)  is the Land\(\acute{e}\)  factor for noble gas nucleus,
\(P^{n}_{0}\)  is the polarization of noble gas nucleus,
\([\rm {N}]\)  is the number density of the noble gas,
and \(\mu =1.07 \times 10^{-26}~\textrm {J/T}\)  is the magnetic moments of \(^3\) He nucleus.
We use Eq. REF  to estimate the \(^3\) He effective magnetic field and numerically calculate the amplification factor.
Based on the demonstrated experimental parameters in ref. [1]}, \(P_{0}^{n} \approx 0.02\)  is the typical polarization of the \(^3\) He gas,
\([\rm {He}] \approx 1.88 \times 10^{26}~\textrm {m}^{-3}\)  corresponds to the number density with 7 amg \(^3\) He.
We obtain \(\lambda M^n P_0^n \approx 2.0~\textrm {mG}\) , which is in good agreement with 2.2 mG demonstrated in ref. [1]}.
According to Eq. REF ,
the amplification factor of \(^3\) He-K spin-based amplifier is estimated to be \(\eta \approx 10^{4}\) .
Compared to \(^{129}\) Xe-Rb systems,
the amplification factor of \(^3\) He-K systems yields about two orders of magnitude improvement.
